All Questions
612 questions with no upvoted or accepted answers
21
votes
0
answers
868
views
Noncommutative arithmetic mean geometric mean inequality and symmetric polynomials
While analyzing convergence speed of stochastic-gradient methods for convex optimization problems, Recht et al (2011) posed a tantalizing conjecture. It seems quite tricky, so after having struggled a ...
- 28.6k
16
votes
0
answers
488
views
An inequality for matrix norms
Working on a problem in combinatorics I come up with the following inequality on matrix norms, which I checked it also numerically:
Let $A=(a_{ij})$ be a real symmetric $n\times n$ matrix with ...
- 4,484
16
votes
1
answer
743
views
Inequalities for marginals of distribution on hyperplane
Let $H = \{ (a,b,c) \in \mathbb{Z}_{\geq 0}^3 : a+b+c=n \}$. If we have a probability distribution on $H$, we can take its marginals onto the $a$, $b$ and $c$ variables and obtain three probability ...
- 156k
16
votes
0
answers
808
views
Determinant inequality involving Hermitian, positive definite matrices
Let $A,B,C\in M_{n}(\mathbb C)$ be Hermitian and positive-definite matrices such that $A+B+C=I_{n}$.
Show that
$$\det\left(6(A^3+B^3+C^3)+I_{n}\right)\ge 5^n\det(A^2+B^2+C^2)$$
This question has been ...
- 269
15
votes
0
answers
749
views
Prove $\int_{0}^{\infty} \cos(\omega x) \exp(-x^{\alpha}) \, {\rm d} x \ge {\alpha^2 \sqrt{\pi} \over 8} \exp \left( -\frac{\omega^2}{4} \right)$
I would like to prove that
$$\int_{0}^{\infty} \cos(\omega x) \exp(-x^{\alpha}) \, {\rm d} x \ge
{\alpha^2 \sqrt{\pi} \over 8} \exp \left( -\frac{\omega^2}{4} \right)$$
for any $\omega > 0$ and $...
- 178
14
votes
0
answers
1k
views
Nice proof of inequality $(1-x^p)^{1/p}(1-x^q)^{1/q}\ge (1-x)(1+x^c)^{1/c}$ where $2^{1/c} = p^{1/p} q^{1/q}$?
Let $0\leq x < 1$, $1 \leq p < \infty$ and $q$ be the conjugate exponent defined by
$$1/p + 1/q = 1.$$
I am looking for a nice proof that
$$ \frac{(1-x^p)^{1/p}(1-x^q)^{1/q}}{(1-x)(1+x^c)^{1/...
- 2,283
14
votes
0
answers
4k
views
Minimum tiling of a rectangle by squares
Given the $n\times m$ rectangle, I want to compute the minimum number of integer-sided squares needed to tile it (possibly of different sizes).
Is there an efficient way to calculate this?
- 1,015
12
votes
0
answers
218
views
Which ordering of factors is needed to obtain this kind of determinantal inequalities?
Let $A$ and $B$ be $n\times n$ Hermitian positive definite matrices. The curious determinantal inequality given here, which can be stated as $$\det (A^{4}+ ABBA+BAAB+B^{4})\ge\det(A^{4}+ AABB+BBAA+B^{...
- 13.4k
11
votes
0
answers
615
views
Monotonicity of ratio of symmetric polynomials
The complete homogeneous symmetric polynomials of degree $\ell$ in $n$ variables are defined by
\begin{equation*}
h_{\ell}(x_1,x_2,\ldots,x_n) = \sum_{1 \leq i_1 \leq i_2 \leq \ldots \leq i_{\ell} \...
11
votes
0
answers
387
views
Inequality for symmetric polynomial functions of log concave variables
Let $(x_i)_{i \ge 1}$ be a log-concave (resp. log-convex) sequence of non-negative real variables. In other words, for $i \ge 2$, we have $x_i^2 \ge x_{i-1}x_{i+1}$ (resp. $x_i^2 \le x_{i-1}x_{i+1}$).
...
- 505
11
votes
0
answers
291
views
$L_2$ minimizing makespan vs. $L_\infty$ minimizing makespan
There are $n$ positive real numbers. We partition these numbers into $m$ parts, the size of each part is the sum the numbers in this part. Maximum size of the parts is called a makespan of a partition....
- 399
10
votes
0
answers
4k
views
Is the conjecture A+B=C following correct?
Is the conjecture on A+B=C following correct ?
Conjecture: Let $A, B, C$ be three positive integer numbers such that $A+B=C$ with $\gcd(A, B, C) = 1$. By Fundamental theorem of arithmetic we write:
$...
- 1,776
10
votes
0
answers
722
views
Fractional Matching version of Hall's Marriage theorem
Let $G=(S,T,E)$ be a bipartite graph, $|S|=|T|$. Then the following are equivalent:
1) there exist a perfect matching in $G$;
2) there exist non-negative weights on edges such that the sum of ...
- 109k
10
votes
0
answers
787
views
Isoperimetric inequality, isodiametric inequality, hyperplane conjecture... what are the inequalities of this kind known or conjectured?
I duplicate here a question I asked on math.stackexchange.
Question: Which inequalities similar to the famous isoperimetric inequality is known?
conjectured?
I recently learned about some ...
9
votes
0
answers
693
views
Van der Pol's identity for the sum of divisors and a quartic polynomial equation for odd perfect numbers
In Touchard (1953) it is mentioned that the sum of divisors $\sigma(n)$ satisfies the following recurrence relation ($n>1$):
$$n^2(n-1) = \frac{6}{\sigma(n)} \sum_{k=1}^{n-1}(3n^2-10k^2)\sigma(k)\...
- 213
9
votes
0
answers
365
views
How to count integer lattice points close to a subspace of $\mathbb R^n$?
Consider $m$ linearly independent vectors in $n$-dimensional Euclidean space, $v_1,...,v_m \in \mathbb R^n$ where $1\leq m<n$, and let $U := {\rm span}(v_1,...,v_m)$ denote the $m$-dimensional ...
- 686
9
votes
0
answers
520
views
Getting a bound via polynomial equations
When studying the existence problem of power residue deference sets, I came across the following system of polynomial equations over $\mathbb{C}$,
\begin{cases}
&\sum\limits_{j=0}^{m-1}x_jx_{2k-j}=...
- 767
8
votes
0
answers
357
views
Bounding a sum of reciprocals of square-free integers
(Cross-posted from MSE, as the question did not get any clear answer)
Fix positive integers $k$ and $n$. Let $N_1,\dots,N_r$ be all the integers less than or equal to $n$ that are squarefree and have ...
- 189
8
votes
0
answers
400
views
When do we have $\|X - Y\| = \|\Sigma(X) - \Sigma(Y)\|$?
For any $X \in \mathbb{C}^{m\times n}$, let $\Sigma(X)$ be the "middle factor" in its SVD, so that $X = U\Sigma(X) V^H$ and the diagonal of $\Sigma(X)$ is arranged in descending order.
...
- 269
8
votes
1
answer
504
views
How large can the dimension of a 'Span of powers of a finite field basis' be?
Let $p$ be a prime. For finite field $\mathbb{F}_{p^k}$ and $d\in\mathbb{Z}^+$, I am considering the following quantity, where we interpret the field $\mathbb{F}_{p^k}$ also as a $\mathbb{F}_p$-vector ...
- 89
8
votes
0
answers
413
views
Eigenvalues of cyclic stochastic matrices
Let's consider the following $n \times n$ cyclic stochastic matrix
$$ M= \begin{pmatrix}
0 & a_2 & & & &b_n \\\
b_1 & 0& a_3& &&& \\\
& b_2 & ...
- 181
8
votes
0
answers
364
views
How localized can a polynomial be in the L1 norm?
Let $0<s<2$ be a parameter, $\Omega = [-1,1]$, and $\Omega_s\subset \Omega$ be a set of measure $s$. I would like to bound the following ratio from above:
$$\sup_{p\in\mathcal{P}_n} \frac{\...
- 3,217
8
votes
0
answers
576
views
A rank inequality
Suppose
$$M := \begin{bmatrix}
M_{11} & \cdots &M_{1d} \\
\vdots & \ddots & \vdots \\
M_{d1} & \cdots & M_{dd}
\end{bmatrix}$$
is a $d \times d$ block matrix such that
$$M_{...
- 500
8
votes
0
answers
491
views
Strange determinant inequality $\det(C+ xA) \det(C-xA) \le (\det C)^2$
Let $A$ be an all-one $3$-by-$3$ matrix, let $C$ be a $3$-by-$3$ matrix, and let $x$ be a real number. How might one prove the following inequality?
$$\det(C+ xA) \det(C-xA) \le (\det C)^2$$
- 99
8
votes
0
answers
1k
views
Infinite Linear Programming
I'm trying to prove optimality for a continuous linear program. That is, I have a linear program with an uncountable number of variables and constraints. I'm not sure how to demonstrate feasibility ...
- 133
7
votes
0
answers
209
views
Li-Yau inequality on $\mathbb R^2$ for functions that are somewhat close to $1$
Let $u:\mathbb R^2\times \mathbb R_{>0}\to \mathbb R_{>0}$ be a positive solution to the heat equation on $\mathbb R^2$ ($u_{xx}+u_{yy}=u_t$, no constants). The Li-Yau inequality in this case ...
- 7,193
7
votes
0
answers
243
views
Loomis-Whitney versus Gagliardo inequalities
When searching for a reference, I discovered a curious fact about the Wikipedia page concerning the Loomis-Whitney Inequality (LWI).This page, which exists only in an English version, states that the ...
- 52.3k
7
votes
0
answers
1k
views
Closed-form solution of a linear programming question
Among all the probability matrices
\begin{equation*}
P =
\left(\begin{array}{cccc}
p_{00} & p_{01} & \ldots & p_{0,J-1} \\
p_{10} & p_{11} & \ldots & p_{1,J-1} \\
\vdots & \...
7
votes
0
answers
317
views
An inequality which involves a sum of integrals
Please help me to prove
$$
\sum\limits_{j=2}^n \frac{1}{j^\alpha (j-1)^\alpha} \int\limits_{j-1}^j \frac{dx}{x^{1-\alpha}(n-x)^\alpha} \leq \int\limits_0^1 \frac{dx}{x^{1-\alpha}(n-x)^\alpha},\quad \...
- 71
7
votes
0
answers
724
views
Inequality between incomplete beta and gamma functions; or when is binomial distribution function above/below its limiting Poisson
Please note: this question was posted first (September 4) in math.stackeschange.com and then (September 16) in stats.stackeschange.com. It got no answers in neither of those sites.
Let the ...
- 337
7
votes
0
answers
486
views
A curious inequality
Let $r_k>0$ for $k = 1,\ldots, n$, let $\alpha_k, \beta_k\in \mathbb{R}$ be given such that
$|\alpha_k|\le \beta_k\le \frac{\pi}{2}$. Suppose further that
$\left|\sum\limits_{k=1}^nr_ke^{i(\...
- 478
6
votes
0
answers
171
views
An inequality involving integer partitions
For integers $n\ge k\ge0$, let $p(n,k)$ denote the number of ways to write $n$ as a sum of $k$ positive integers (repetition allowed). For example, $p(6,3)=3$ since
$$6=1+1+4=1+2+3=2+2+2.$$
QUESTION. ...
- 15.6k
6
votes
0
answers
129
views
Weak-type inequality for the partial Fourier sum operator
I'm studying harmonic analysis by myself. One of the online notes gives the following claim as a remark:
For any $N \in \mathbb{Z}^{+}$, let's use $S_{N}$ to denote the partial ($N$ terms) Fourier sum ...
- 349
6
votes
1
answer
489
views
What inequalities for convex sets are known since the work of Scott and Awyong?
In 2000, Paul R. Scott and Poh Way Awyong published the paper Inequalities for Convex Sets, which nicely collates the known results relating various natural geometric functionals (diameter, area, etc.)...
- 3,068
6
votes
0
answers
381
views
An inequality related to the numbers of faces of polytopes with d+2 facets
I would like to prove an inequality related to the number of $k$-faces of two $d$-polytopes with $d+2$ facets; see (1) below.
Let $r>0$, $s>0$, $t\ge 0$, and $d\ge 2$ be such that $d=r+s+t$. We ...
6
votes
0
answers
132
views
Q-analogue of an inequality
Pick integers $b\geq a \geq 0$ and $k\geq j\geq 0$.
It is not super-difficult to prove the inequality
$$
\binom{kb}{ka}^j \geq \binom{jb}{ja}^k.
$$
This is actually quite a nice inequality that was ...
- 15.8k
6
votes
0
answers
129
views
A reference for an integrability property?
In a recent paper of mine (Compensated integrability), I established a functional inequality which has nice consequences. For instance, it contains the isoperimetric inequality, and it gives a new ...
- 52.3k
6
votes
0
answers
274
views
An inequality in cyclic polygon and tangential polygon
I proposed my conjecture, it is strengthened version of the Erdős–Mordell inequality as following:
Let $A_1A_2.....A_n$ be a cyclic polygon and $B_1B_2....B_n$ be the its tangential polygon. Let $P$ ...
- 1,044
6
votes
0
answers
587
views
Lower bound on the sum of singular values for a sum of Hermitian matrices
Denote the eigenvalues of an $n\times n$ matrix $\mathbf{X}$ by $\lambda_i(\mathbf{X})$ and its singular values by $\sigma_i(\mathbf{X})$, $i=1,\ldots,n$. When $\mathbf{X}$ is Hermitian, we know that $...
- 907
6
votes
0
answers
97
views
Finding the optimal mixture of two convex functions
I am trying to find an efficient way to solve the problem $$\min_{p,x_1,x_2} p\cdot f(x_1)+ (1-p) \cdot f(x_2)~~~~~ s.t.\\p\cdot g_1(x_1) + (1-p)\cdot g_2(x_2)\leq 1 \\ 0\leq p \leq 1$$ where $x_1,x_2\...
6
votes
0
answers
243
views
Operator arithmetic-harmonic mean inequality with operator-valued weights
Let $\Lambda_1,\dots,\Lambda_n$ be strictly positive definite operators in the Euclidean space $\mathbb{R}^d$. By an operator arithmetic-harmonic mean inequality with weights $\Lambda_i$ I mean the ...
- 4,000
6
votes
0
answers
336
views
Maximal inequalities for square of partial sums
Let $S_n = \sum_{i \leq n} X_i$ be the partial sums of a nice sequence of random variables $X_i$. In my application, $X_i$ is a functional of a finite-state, irreducible, aperiodic Markov chain, so ...
- 1,069
6
votes
0
answers
342
views
Maximizing Renyi entropy for a certain channel
The channel under consideration is $T = A + B$, where $A$ and $B$ take on values in $\{0, 1\}$ according to a probability mass function. Let (joint) random vector $(A_1, A_2,\ldots, A_n)$ be denoted ...
6
votes
0
answers
317
views
Variant of orthogonal Procrustes problem
The orthogonal Procrustes problem seeks a matrix $M$ that minimizes $||AM-B||_F$ subject to $M^TM=I$, where $M$ is $d\times d$ and both $A$ and $B$ are $n\times d$. Geometrically, $M$ rotates a set of ...
- 61
6
votes
0
answers
630
views
Counting permutation matrices in 0,1,2 matrices
Let $M$ be a matrix with entries equal to 0, 1 or 2, such that all of the row and column sums are equal to $c$.
The Van der Waerden bound gives (roughly) the following bound on the permanent of $M$:
...
- 1,633
5
votes
0
answers
204
views
A proof for an $L^p$-$L^p$ inequality
This is a transfer of the question
https://math.stackexchange.com/questions/4996853/an-lp-lp-inequality
Let $a\in (0,1)$ and $1<p<\infty$ and use $L^{p}$ to denote the space $L^{p}([0,\infty))$ ...
- 852
5
votes
0
answers
183
views
On the polynomials $\sum_{k=0}^n\binom{n+k}k^m q^k$
A sequence of polynomials
$$P_0(q),\ P_1(q),\ P_2(q),\ \ldots$$
with real coefficients is called $q$-log-convex if for each $n=1,2,3,\ldots$ every coefficient of the polynomial $P_{n+1}(q)P_{n-1}(q)-...
- 15.6k
5
votes
0
answers
167
views
Bounding elementary symmetric polynomials away from zero
Let $2 \leq m \leq n$ be integers and let $\mathbf{x} \in \mathbb{R}^n$ (importantly, I am not assuming that the entries of $\mathbf{x}$ are non-negative). The elementary symmetric polynomials are ...
- 6,351
5
votes
0
answers
169
views
Is there a sharper Golden–Thompson inequality?
For any two Hermitian matrices $A$ and $B$, the Golden–Thompson inequality
$$\mathrm{Tr} (e^A e^B) \geq \mathrm{Tr} \, e^{A + B}$$
holds, and it is known to be a strict inequality whenever $[A, B] \...
- 51
5
votes
0
answers
415
views
Extending Gromov's inequality
In 1981 Gromov proved that all Riemannian metrics on the complex projective space $\mathbb CP^n$ satisfy the bound
$$\DeclareMathOperator{stsys}{stsys} \DeclareMathOperator{vol}{vol}
\frac{\stsys_2^n}{...
- 16.6k